The blue area (weight painting) still moves when a bone is moved. Why the hell does this happen?
Because some vertices may appear blue but it doesn't mean there are not part of the vertex group, with an influence of 0.01 they will appear blue. If you want to have a real preview you need to switch the Viewport Overlay > Weight Paint > Zero Weight from None to Active. Now black is real 0%.



If you want to make sure that some parts of your mesh are not part of a vertex group, select these vertices in Edit mode and in the Properties panel > Object Data > Vertex Group, select the group and click on Remove or Remove From All Groups.